.. _use-case-stressed-beam: A simple stressed beam ====================== We consider a simple beam stressed by a traction load F at both sides. .. figure:: ../_static/axial-stressed-beam.png :align: center :alt: use case geometry :width: 50% Beam geometry The geometry is supposed to be deterministic; the diameter D is equal to: .. math:: D=0.02 \textrm{ (m)} By definition, the yield stress is the load divided by the surface. Since the surface is :math:`\pi D^2/4`, the stress is: .. math:: S=\frac{F}{ \pi D^2/4} Failure occurs when the beam plastifies, i.e. when the axial stress gets larger than the yield stress: .. math:: R - \frac{F}{ \pi D^2/4} \leq 0 where :math:`R` is the strength. Therefore, the limit state function :math:`G` is: .. math:: G(R,F) = R - \frac{F}{\pi D^2/4}, for any :math:`R,F \in \mathbb{R}`. The value of the parameter :math:`D` is such that: .. math:: D^2/4 = 10^{-4}, which leads to the equation: .. math:: G(R,F) = R - \frac{F}{10^{-4} \pi}. We consider the following distribution functions. ======== ================================================================================ Variable Distribution ======== ================================================================================ R LogNormal( :math:`\mu_R= 3 \times 10^6`, :math:`\sigma_R=3 \times 10^5` ) [Pa] F Normal( :math:`\mu_F=750` , :math:`\sigma_F=50`) [N] ======== ================================================================================ where :math:`\mu_R=E(R)` and :math:`\sigma_R^2=V(R)` are the mean and the variance of :math:`R`. The failure probability is: .. math:: P_f = \Prob{G(R,F) \leq 0}. The exact :math:`P_f` is .. math:: P_f = 0.02920. API documentation ----------------- .. currentmodule:: openturns.usecases.stressed_beam .. autoclass:: AxialStressedBeam :noindex: Examples based on this use case ------------------------------- .. minigallery:: openturns.usecases.stressed_beam.AxialStressedBeam